Achilles is said to have dragged the corpse of Hector through the dust behind his chariot -- did the Greeks have any other use for chariots beyond elaborate gloating, and how did those uses change with time? References for the Greeks and Romans' employment of chariots would be most welcome!
The Iliad itself describes chariots used to transport heroes to and from the battlefield; only seldom are the chariots involved in direct combat (usually when someone like the driver is struck down by a spear or when a hero hurls a spear from a chariot). This also appears to have been the way chariots were used in Late Bronze Age Greece, as far as we can tell, as well as during the Early Iron Age/Archaic period, judging by the iconographic sources.
My answers in these threads are pertinent:
And I should stress that Homer doesn't know anything about the Late Bronze Age; he's firmly rooted in his own time (ca. 700 BC or even a little later), so what he describes about chariots seems to be contemporary practice; it just so happens that the Mycenaean Greeks deployed the chariots in a similar way. (For details, see this article.)
Since much of our interpretation of chariots in the Aegean/Greece rests on interpreting the iconographic evidence, you may also want to check out the answer I gave (with help from others) here. That answer was recently published as a slightly expanded article on Ancient World Magazine.
As far as the Romans are concerned, I'm not an expert, so I'll leave that to others.